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Let $L$ be a $2 \times 2$ matrix. For each of the following give a proof or counterexample.

a) If $L^{2}=0$ then $L=0$.
b) If $L^{2}=L$ then either $L=0$ or $L=I$.
c) If $L^{2}=I$ then either $L=I$ or $L=-I$.
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